What is regression layer in a spatial transformer?
Basically, it means that the "localization network" should output a set of real valued parameters (typically 6 numbers). The word "regression" doesn't bear any specific meaning. 

Any network that relies on the original image as input (directly or indirectly) and outputs 6 numbers, would work. And its last layer would qualify as "regression layer" as long as it is not restricted in the real domain (not normalized, softmaxed, etc)